Smoothness of convolution products of orbital measures on rank one compact symmetric spaces

TL;DR

This paper investigates the smoothness properties of convolution products of orbital measures on rank one compact symmetric spaces, establishing conditions for absolute continuity and $L^{2}$ integrability based on double coset dimensions.

Contribution

It characterizes when convolution products of orbital measures are absolutely continuous or in $L^{2}$ on rank one symmetric spaces, including explicit results for $SU(2)/SO(2)$.

Findings

All convolution products of pairs of continuous orbital measures are absolutely continuous.

Convolution products are in $L^{2}$ except in the case of $SU(2)/SO(2)$.

Pairs with convolution in $L^{2}$ are characterized by double coset dimensions.

Abstract

We prove that all convolution products of pairs of continuous orbital measures in rank one, compact symmetric spaces are absolutely continuous and determine which convolution products are in $L^{2}$ (meaning, their density function is in $L^{2})$. Characterizations of the pairs whose convolution product is either absolutely continuous or in $L^2$ are given in terms of the dimensions of the corresponding double cosets. In particular, we prove that if $G/K$ is not $SU(2)/SO(2),$ then the convolution of any two regular orbital measures is in $L^{2}$, while in $SU(2)/SO(2)$ there are no pairs of orbital measures whose convolution product is in $L^{2}$.